Earlier quoted context omitted.
Does anyone know how to learn more about one's DNA for the purpose OP suggested, in a way that respects privacy?
“In a way that respects privacy” is very difficult. There are companies (Dante Labs for instance), which will do comprehensive genetic testing (full genome sequencing), but privacy is complex. Once the data exists, it’s very hard to un-exist it.
I would highly recommend avoiding Dante Labs. I had a terrible experience with them. There were extreme delays, lasting many months. When I requested the full genome be sent to me, they insisted the only way to do it was a hard drive. This hard drive was sent from Italy to California. On its way, it was seized by customs in Mumbai, who opened it and re-sealed it. The package arrived in tatters, and based on my analysis of the genome with the appropriate reference, it appeared to be someone else's genome. Many of the SNPs disagreed with a later 23andme report I got, which I assume is more accurate.